We observe from this plot that the true distribution of the standardized von Neumann entropy \(X\)  is non-symmetric and appears to be left-skewed when \(m=2\) , \(n=4\) . In comparison, when the subsystem dimensions increase to \(m=16\) , \(n=32\)  with the same ratio \(m/n=1/2\) , the distribution of \(X\)  appears to be closer to the Gaussian distribution. The observed asymptotic Gaussian behavior is typical for a wide class of linear spectral statistics (such as the von Neumann entropy) over different
random matrix ensembles as already been observed in the Hilbert-Schmidt ensemble [1]} and Bures-Hall ensemble [2]}. Here, we also conjecture that in the limit
\(m\rightarrow \infty ,~~~~n\rightarrow \infty ,~~~~\frac{m}{n}\in (0,1],\) 
